The plausible relativistic physical variables describing a
spinning, charged and massive particle are, besides the charge
itself, its Minkowski (four) po­ sition X, its relativistic linear
(four) momentum P and also its so-called Lorentz (four) angular
momentum E # 0, the latter forming four trans­ lation invariant
part of its total angular (four) momentum M. Expressing these
variables in terms of Poincare covariant real valued functions
defined on an extended relativistic phase space [2, 7J means that
the mutual Pois­ son bracket relations among the total angular
momentum functions Mab and the linear momentum functions pa have to
represent the commutation relations of the Poincare algebra. On any
such an extended relativistic phase space, as shown by Zakrzewski
[2, 7], the (natural?) Poisson bracket relations (1. 1) imply that
for the splitting of the total angular momentum into its orbital
and its spin part (1. 2) one necessarily obtains (1. 3) On the
other hand it is always possible to shift (translate) the commuting
(see (1. 1)) four position xa by a four vector ~Xa (1. 4) so that
the total angular four momentum splits instead into a new orbital
and a new (Pauli-Lubanski) spin part (1. 5) in such a way that (1.
6) However, as proved by Zakrzewski [2, 7J, the so-defined new
shifted four a position functions X must fulfill the following
Poisson bracket relations: (1.

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